There's no issue with «`» (within single quotes). The problem is with «'».

Ways of making «foo$'bar$`baz» into a bash literal:

foo\$\'bar\$\`baz
"foo\$'bar\$`baz"
'foo$'"'"'bar$`baz'
'foo$'\''bar$`baz'
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Of course, you could use avoid using «'», starting by writing code that's much clearer.

perl -lanE'$F[1]=~/(.*?)(chr\w[^_]?)_.*\b(.*)/; say "$1\t$1$2$3\t$3"'
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The pattern looks very fishy. /.*\b.*/? There are clearer and safer ways of writing that. I have no idea what you are trying to match with that.